Modelowanie Nanostruktur - University of Warsajmajewsk/MODNanoSTR_L6.pdf · Modelowanie Nanostruktur Lecture 6 1 Modelowanie Nanostruktur Semester Zimowy 2011/2012 Wykład Jacek A

Modelowanie Nanostruktur

Lecture 6 1


Semester Zimowy 2011/2012

Wykład

Jacek A. Majewski

Chair of Condensed Matter Physics

Institute of Theoretical Physics

Faculty of Physics, Universityof Warsaw

E-mail: [email protected]

Struktura elektronowa nanorurek Zwiazki wegla Struktura elektronowa grafenu Od grafenu do nanorurki w przestrzeni prostej i odwrotnej

Modelowanie Nanostruktur, 2011/2012

Jacek A. Majewski

Wykład 6 – 15 XI 2011

[email protected]

Carbon Compounds –

Diamonds of the 21st century

Electronic structure of graphene

Carbon nanotubes (CNTs) – geometry, properties,

& applications

Electronic structure of carbon nanotubes (CNT)

CNT & graphene based electronics –

the future of information technologies ?

1. diamond

2. graphite

3. fullerene

4. graphene

5. carbon nanotubes

6. carbon nanocoils

7. lonsdaleite "hexagonal diamond"

8. amorphous carbon

9. carbon nanofoam

10. .....

Allotropes of carbon


Lecture 6 2

Covalent bonds between carbons

sp3 and sp2 hybrids

Diamond

Lattice constant

0.3566 nm at 298 K.

nearest neighbor distance:

0.154450 nm at 298K.

Atomic weight: 12.01

Atomic radius: 0.077 nm

Number of atoms in a

unit cell: 8

Two fcc lattices

shifted by (a/4) [111] The hardest material !

Graphite

STM image

projection of

a = b= 0.2456 nm, c= 0.6694 nm

The carbon-carbon bond length in the bulk

form is 0.1418 nm (shorter and stronger

than in diamond)

The interlayer spacing is c/2 = 0.3347 nm

weakly coupled 2D planes

pencil, lubricant

Fullerenes

The C60 cluster

‘buckminsterfullerene’

‘bucky-ball’

60 carbon atoms formed in

12 pentagons

20 hexagons

diameter = 1.034 nm

Point group –

120 symmetry operations

Synthesized by R. F. Curl,

H. W. Kroto, and R. E. Smalley

Nobel Prize for Chemistry 1996

Named after Buckminster Fuller

American architect

(living XIX-XX century)


Lecture 6 3

Fullerenes

C20

consists of 12 pentagons

ideal of dodecahedron

C40

Fullerenes

C86 C540

and many more …. (up to C980 )

Carbon nanotubes (CNTs)

S. Iijima, Nature 354, 56 (1991)

D. Vgarte, Nature 359, 707 (1992)

Multiple Wall Carbon Nanotubes


Lecture 6 4

Boron Nitride Nanotubes

Carbon (Boron Nitride) Nanocoils

„These nanotubes are so beautiful that

they must be useful for something”

R. Smalley

CNTs – Mechanical Properties

Mechanical strength – graphite-like strong bonds

-- no dangling bonds

-- no weakly bound sheets


Lecture 6 5

Graphene: a sheet of carbon atoms

What is graphene?

2-dimensional

hexagonal lattice

of carbon

sp2 hybridized

carbon atoms

Among strongest

bonds in nature

Basis for: C-60 (bucky balls) nanotubes

graphite

Graphene – a single sheet of C atoms

x

y Two unit-cell vectors:

Two non-equivalent

atoms A and B in the unit cell

(two sublattices)

a a( , )1

3 1

2 2

a a( , ) 2

3 1

2 2

M. Machon, et al., Phys. Rev. B 66, 155410 (2002)

The band structure

was calculated with

a first-principles

method

Electronic band structure of graphene

Γ Q Q P

Γ

P Q

xk

yk

Brillouin Zone


Lecture 6 6

Tight-binding description of graphene

σ bonds – not considered

in this model

π bonds considered

Only couplings between

nearest neighbors taken into

account

One pz orbital pro atom

Tight-binding description of graphene

p AB

*AB p

ε ε( k ) H ( k )ε( k )

H ( k ) ε ε( k )

0

AB n A A B B n

Rn

ˆH ( k ) exp( ik R ) φ ( τ ) | H | φ ( τ R )

ABH ( k ) t [ exp( ik a ) exp( ik a )] 1 21

pε 0

*AB AB

/ε( k ) t H ( k )H ( k )

1 2

(zero of energy)

AA BB pH H ε

0p

t

0

p t

Dispersion relations for graphene

y yx

/k a k ak a

ε( k ) t cos cos cos

2

1 23

1 4 42 2 2

Nearest-neighbors tight-binding

electronic structure of graphene

T-B

Ab-initio

Γ Q Q P

Γ

P Q

xk

yk

Brillouin Zone

t = -2.7 eV

Hopping parameter


Lecture 6 7

Tight-binding band structure of graphene

y yx

/k a k ak a

ε( k ) t cos cos cos

2

1 23

1 4 42 2 2

Graphene is

semi-metallic

Energy gap is

equal zero only

in one k-point

(P-point)

Massless 2D Dirac Fermions

“light cone”

Reciprocal lattice of graphene

Carbon nanotubes: geometry & electronic structure


Lecture 6 8

Nanotube = rolled graphene sheet

hC ( n,m ) na ma 1 2

Nanotube is specified by

chiral vector:

hC ( n,m ) na ma 1 2

Structure of Carbon Nanotubes

The chiral vector:

m n

(n,0) – zig-zag

(n,n) – armchair

(n,m) – chiral

m n

Structure of Carbon Nanotubes

Carbon nanotube

of type chair (5,5)

Carbon nanotube

of type zigzag (9,0)

Chiral (10,5)

carbon nanotube

(8,4) chiral tube (7,0) zig-zag tube (7,7) armchair tube

Perspective view of nanotubes


Lecture 6 9

Electronic structure of CNT –

Zone-folding Approximation

Graphene – infinite plane in 2D

For CNTs, we have a structure which is

macroscopic along the tube direction,

but the circumference is in atomic scale

Periodic boundary conditions

in the circumferential direction

The allowed electronic states are restricted

to k-vectors that fulfill the condition

hk C ( n,m ) πl 2

P-point belongs to allowed k-vectors CNT is metallic

P-point does not belongs to allowed k-vectors

CNT is semiconducting

Which CNTs are metallic?

Which semiconducting?

GAP ABE H ( k ) 0 0

GAPE exp( ik a ) exp( ik a ) 1 20 1 0

We get two possible conditions

k a π l

1

12

3k a π l'

2

22

3and

or

k a π l

1

22

3k a π l'

2

12

3and

Due to the periodicity condition

hk C ( n,m ) k ( na ma ) πl'' 1 2 2

πn l πm l' πl''

1 22 2 2

3 3

n ml

2

3

integersl ,l',l''

n ml'

2

3

or

πn l πm l' πl''

2 12 2 2

3 3

Which CNTs are metallic?

Which semiconducting?

n ml

2

3n m

l'

2

3

Nanotube (n,m)

is metallic n m l 3

Nanotube is a metal if n-m is multiple of three

Otherwise CNT is a semiconductor

All armchair (n=m) CNTs are metallic


Lecture 6 10

Metallic and semi-conducting CNTs

Metallic and semi-conducting CNTs

Metallic and semi-conducting CNTs – Band Structure

CNTs – band structure & density of states


Lecture 6 11

Electronic density of states for

(16,0), (13,6), (21,20) nanotubes

Pronounced 1D-behavior !

CNTs – Ideal 1D Quantum Wires

Transverse momentum quantization:

is only allowed mode,

all others more than 1eV away (ignorable bands)

1D quantum wire with two spin-degenerate transport

channels (bands)

Massless1D Dirac Hamiltonian

Two different momenta for backscattering

CNT & graphene based FETs - the future of nanoelectronics?

Molecular Electronics with fullerenes and

Transistors based on CNTs


Lecture 6 12

TIME

Scale

10 nm?

2015 ?

Electronics based on semiconductor

nanostructures and large molecules

“Top down”

“Bottom up”

?

?

Semiconductor nano-wires

& carbon nanotubes

Field Effect Transistor based

on silicon nano-wire & carbon nanotube

CNTFETs: 1998 - 2004

Back gate transistor Top gate transistor

CNTFETs – Isolated Top Gate Devices

Schematic cross section of

a top gate CNTFET

Output characteristics of

a p-type device with Ti gate

and a gate oxide thickness

of 15 nm.

S. J. Wind et al.,

Appl. Phys. Lett. 80, 3817 (2002)


Lecture 6 13

Comparison of Si-MOSFETs with up-scaled

CNT-MOSFETs

CNTs devices show competitiveness to state-of-the-art

Si-MOSFETs !

CNT-MOSFET shows unprecedented values for

transconductance and maximum current drive

Integrated circuit built on single nanotube

Ring oscillator circuit built on a

single carbon nanotube

consisting of five CMOS inverter

stages.

Nanotube covered by the

contact and gate electrodes.

IBM T. J. Watson Research

Center, the University of Florida,

and Columbia University

[Chen et al., Science(2006) 311,

1735].

Integrated circuit built on single nanotube

Metals with different work functions as the gates

Al Pd

Gate Gate

p-type FET n-type FET

SWCNT

The difference in the

two work functions

shifts the characteristics

to give a p-/ n-FET pair

In this way, five inverters involving ten FETs were arranged

side-by-side on a single, 1.8 µm long SWNT.

Inverter works at a frequency of 52 MHz, ~100 000 times

faster than previous circuits built by connecting

separate nanotube transistors.

This improvement is a result of our compact design, which

eliminates parasitic capacitance contributions to a large extent

+ = Inverter

Graphene for devices


Lecture 6 14

Graphene’s advantage: cut-a-structure

Graphene Nanoribbon FETs (GNR FETs)

I-V characteristics for

different GNR widths The schematic sketch

of an GNR

Scheme of GNR FET I-V characteristics for n=12 GNRs

with charge impurities

On May 21, 2009,, HRL laboratories said that it had

made devices from single-layer graphene on 2 inch

diameter 6H-SiC wafers with much-improved

performance figures.

Epitaxial graphene based devices

“They have world-record field

mobility of approximately

6000 cm2/Vs, which is six to

eight times higher than current

state-of-the-art silicon n-

MOSFETs,”

IEEE Electron Devices Lett. 30, 650-652 (2009)

Summary

Fascinating world of carbon compounds

Are carbon compounds based devices

the future of information technologies?

Jacek A. Majewski, University of Warsaw

Electronic structure of graphene & CNTs

It’s not clear yet, but

Carbon compounds definitely changed

the way of thinking about materials science


Lecture 6 15

?

When it happens?

Thank you

Documents

Modelowanie Nanostruktur - University of Warsajmajewsk/MODNanoSTR_L6.pdf · Modelowanie Nanostruktur Lecture 6 1 Modelowanie Nanostruktur Semester Zimowy 2011/2012 Wykład Jacek A